Blood flow

Blood flow is the continuous circulation of blood in the cardiovascular system. This process ensures the transportation of nutrients, hormones, metabolic wastes, O₂ and CO₂ throughout the body to maintain cell-level metabolism, the regulation of the pH, osmotic pressure and temperature of the whole body, and the protection from microbial and mechanical harms.^[1]

The science dedicated to describe the physics of blood flow is called hemodynamics. For the basic understanding it is important to be familiar with the anatomy of the cardiovascular system and hydrodynamics. However it is crucial to note that blood is a non-Newtonian fluid, best studied using rheology and not hydrodynamics. Blood vessels are not rigid tubes, so classic hydrodynamics and fluids mechanics based on the use of classical viscometers are not capable of explaining hemodynamics.^[2]

Blood and its composition

Blood is a complex liquid. Blood is composed of plasma and formed elements. The plasma contains 91.5% water, 7% proteins and 1.5% other solutes. The formed elements are platelets, white blood cells and red blood cells, the presence of these formed elements and their interaction with plasma molecules are the main reasons why blood differs so much from ideal Newtonian fluids.^[3]

Mechanics of blood circulation

Mechanics is the study of motion (or equilibrium) and the forces that causes it. Blood moves in the blood vessels, while the heart serves as the pump for the blood. The vessel walls of the heart are elastic and movable, enabling the blood and the wall to exert forces on each other that in turn influence their respective motion. Therefore, to understand the mechanics of blood circulation, it is worthwhile to review the basic mechanics of fluid, the elastic solids (momentum), and the nature of the forces exerted between two moving substances in contact.

Velocity

Often expressed in cm/s. This value is inversely related to the total cross-sectional area of the blood vessel and also differs per cross-section, because in normal condition the blood flow has laminar characteristics. For this reason the blood flow velocity is the fastest in the middle of the vessel and slowest at the vessel wall. In most cases the mean velocity is used.^[4] There are many ways to measure blood flow velocity, like videocapillary microscoping with frame-to-frame analysis, or laser Doppler anemometry.^[5] Blood velocities in arteries are higher during systole than during diastole. One parameter to quantify this difference is the pulsatility index (PI), which is equal to the difference between the peak systolic velocity and the minimum diastolic velocity divided by the mean velocity during the cardiac cycle. This value decreases with distance from the heart.^[6]

$PI = \frac{v_{systole} - v_{diastole}}{v_{mean}}$

Relation between blood flow velocity and total cross-section area in human^[7]

Type of blood vessels	Total cross-section area	Blood velocity in cm/s
Aorta	3–5 cm^2	40 cm/s
Capillaries	4500–6000 cm^2	0.03 cm/s[8]
Vena cavae inferior and superior	14 cm^2	15 cm/s

Stress

When force is applied to a material it starts to deform or move. As the force needed to deform a material (e.g. to make a fluid flow) increases with the size of the surface of the material A.,^[9] the magnitude of this force F is proportional to the area A of the portion of the surface. Therefore, the quantity (F/A) that is the force per unit area is called the stress. The shear stress at the wall that is associated with blood flow through an artery depends on the artery size and geometry and can range between 0.5 to 4 Pa.^[10]

$\sigma = \frac{F}{A}$.

Under normal conditions, to avoid atherogenesis, thrombosis, smooth muscle proliferation and endothelial apoptosis, shear stress maintains its magnitude and direction within an acceptable range. In some cases occurring due to blood hammer, shear stress reaches larger values. While the direction of the stress may also change by the reverse flow, depending on the hemodynamic conditions. Therefore, this situation can lead to atherosclerosis disease.^[11]

Laminar shear of fluid between two plates. $v=u, \tau=\sigma$. Friction between the fluid and the moving boundaries causes the fluid to shear (flow). The force required for this action per unit area is the stress. The relation between the stress (force) and the shear rate (flow velocity) determines the viscosity.

Viscosity of plasma

Normal blood plasma behaves like a Newtonian fluid at physiological rates of shear. Typical values for the viscosity of normal human plasma at 37 °C is 1.4 mN·s/m².^[12] The viscosity of normal plasma varies with temperature in the same way as does that of its solvent water; a 5 °C increase of temperature in the physiological range reduces plasma viscosity by about 10%.

Osmotic pressure of plasma

The osmotic pressure of solution is determined by the number of particles present and by the temperature. For example, a 1 molar solution of a substance contains 6.022×10²³ molecules per liter of that substance and at 0 °C it has an osmotic pressure of 2.27 MPa (22.4 atm). The osmotic pressure of the plasma affects the mechanics of the circulation in several ways. An alteration of the osmotic pressure difference across the membrane of a blood cell causes a shift of water and a change of cell volume. The changes in shape and flexibility affect the mechanical properties of whole blood. A change in plasma osmotic pressure alters the hematocrit, that is, the volume concentration of red cells in the whole blood by redistributing water between the intravascular and extravascular spaces. This in turn affects the mechanics of the whole blood.^[9]

Red blood cells

The red blood cell is highly flexible and biconcave in shape. Its membrane has a Young's modulus in the region of 106 Pa. Deformation in red blood cells is induced by the shear stress. When a suspension is sheared, the red blood cells deform and spin because of the velocity gradient, with the rate of deformation and spin depending on the shear-rate and the concentration. This can influence the mechanics of the circulation and may complicate the measurement of blood viscosity. It is true that in a steady state flow of a viscous fluid through a rigid spherical body immersed in the fluid, where we assume the inertia is negligible in such a flow, it is believed that the downward gravitational force of the particle is balanced by the viscous drag force. From this force balance the speed of fall can be shown to be given by Stokes' law

$U_s = \frac{2}{9}\frac{\left(\rho_p - \rho_f\right)}{\mu} g\, a^2$^[9]

Where a is the particle radius, ρ_p, ρ_f are the respectively particle and fluid density μ is the fluid viscosity, g is the gravitational acceleration. From the above equation we can see that the sedimentation velocity of the particle depends on the square of the radius. If the particle is released from rest in the fluid, its sedimentation velocity U_s increases until it attains the steady value called the terminal velocity (U), as shown above.

We have looked at blood flow and blood composition. Before we look at the main issue, hemodilution, let us take a brief history into the use of blood. Its therapeutic use is not a modern phenomenon. Egyptian writings dates back at least 2000 years suggest oral ingestion of blood as a ‘sovereign remedy’ for leprosy. Experiments with the first intravenous blood transfusions began at the start of the 16th century, and in the last 50 years the field of transfusion medicine has progressed remarkably, bringing with it an increase in the use of blood and blood products.^[13] However, the therapeutic use of blood comes with significant risks. As a result, many persons are searching for alternatives to the transfusion of whole blood. Today, bloodless medicine and surgery (BMS) programs have been developed not only for people with certain religious beliefs, but also for patients who fear the risks of blood transfusions and desire to take the best possible medical precautions.

Hemodilution

Hemodilution is the dilution of the concentration of red blood cells and plasma constituents by partially substituting the blood with colloids or crystalloids. It is a strategy to avoid exposure of patients to the hazards of homologous blood transfusions.

Hemodilution can be normovolemia which, as we said, implies the dilution of normal blood constituents by the use of expanders. During acute normovolemic hemodilution (ANH) blood subsequently lost during surgery contains proportionally fewer red blood cells per millimetre, thus minimizing intraoperative loss of the whole blood. Therefore, blood lost by the patient during surgery is not actually lost by the patient, for this volume is purified and redirected into the patient.

There is however hypervolemic hemodilution (HVH). Here, instead of simultaneously exchanging the patient’s blood as in ANH, the hypervolemic technique is carried out by using acute preoperative volume expansion without any blood removal. In choosing a fluid, however, it must be assured that when mixed the remaining blood behaves in the microcirculation as in the original blood fluid, retaining all its properties of viscosity.^[14]

In presenting what volume of ANH should be applied one study suggests a mathematical model of ANH which calculates the maximum possible RCM savings using ANH, given the patients weight H_i and H_m. Not to worry. Attached to this document is a glossary of the term used.

To maintain the normovolemia, the withdrawal of autologous blood must be simultaneously replaced by a suitable hemodilute. Ideally, this is achieved by isovolemia exchange transfusion of a plasma substitute with a colloid osmotic pressure (OP). A colloid is a fluid containing particles that are large enough to exert an oncotic pressure across the micro-vascular membrane. When debating the use of colloid or crystalloid, it is imperative to think about all the components of the starling equation:

$\ Q = K ( [P_c - P_i]S - [P_c - P_i] )$

To identify the minimum safe hematocrit desirable for a given patient the following equation is useful:

$\ BL_s = EBV \ln \frac{H_i}{H_m}$

where EBV is the estimated blood volume; 70 mL/kg was used in this model and H_i (initial hematocrit) is the patient’s initial hematocrit. From the equation above it is clear that the volume of blood removed during the ANH to the H_m is the same as the BL_s. How much blood is to be removed is usually based on the weight, not the volume. The number of units that need to be removed to hemodilute to the maximum safe hematocrit (ANH) can be found by

$ANH = \frac {BL_s}{450}$

This is based on the assumption that each unit removed by hemodilution has a volume of 450 mL (the actual volume of a unit will vary somewhat since completion of collection ais dependent on weight and not volume). The model assumes that the hemodilute value is equal to the H_m prior to surgery, therefore, the re-transfusion of blood obtained by hemodilution must begin when SBL begins. The RCM available for retransfusion after ANH (RCMm) can be calculated from the patient's H_i and the final hematocrit after hemodilution(H_m)

$RCM = EVB \times (H_i - H_m)$

The maximum SBL that is possible when ANH is used without falling below Hm(BLH) is found by assuming that all the blood removed during ANH is returned to the patient at a rate sufficient to maintain the hematocrit at the minimum safe level

$BL_H = \frac {RCM_H} {H_m}$

If ANH is used as long as SBL does not exceed BL_H there will not be any need for blood transfusion. We can conclude from the foregoing that H should therefore not exceed s. The difference between the BL_H and the BL_s therefore is the incremental surgical blood loss (BL_i) possible when using ANH.

$\ {BL_i} = {BL_H} - {BL_s}$

When expressed in terms of the RCM

${RCM_i} = {BL_i} \times {H_m}$

Where RCM_i is the red cell mass that would have to be administered using homologous blood to maintain the H_m if ANH is not used and blood loss equals BLH.

The model used assumes ANH used for a 70 kg patient with an estimated blood volume of 70 ml/kg (4900 ml). A range of H_i and H_m was evaluated to understand conditions where hemodilution is necessary to benefit the patient.^[15][16]

Result

The result of the model calculations are presented in a table given in the appendix for a range of H_i from 0.30 to 0.50 with ANH performed to minimum hematocrits from 0.30 to 0.15. Given a H_i of 0.40, if the H_m is assumed to be 0.25.then from the equation above the RCM count is still high and ANH is not necessary, if BL_s does not exceed 2303 ml, since the hemotocrit will not fall below H_m, although five units of blood must be removed during hemodilution. Under these conditions, to achieve the maximum benefit from the technique if ANH is used, no homologous blood will be required to maintain the H_m if blood loss does not exceed 2940 ml. In such a case ANH can save a maximum of 1.1 packed red blood cell unit equivalent, and homologous blood transfusion is necessary to maintain H_m, even if ANH is used. This model can be used to identify when ANH may be used for a given patient and the degree of ANH necessary to maximize that benefit.

For example, if H_i is 0.30 or less it is not possible to save a red cell mass equivalent to two units of homologous PRBC even if the patient is hemodiluted to an H_m of 0.15. That is because from the RCM equation the patient RCM falls short from the equation giving above. If H_i is 0.40 one must remove at least 7.5 units of blood during ANH, resulting in an H_m of 0.20 to save two units equivalence. Clearly, the greater the H_i and the greater the amount of units removed during hemodilution, the more effective ANH is for preventing homologous blood transfusion. The model here is designed to allow doctors to determine where ANH may be beneficial for a patient based on their knowledge of the H_i, the potential for SBL, and an estimate of the H_m. Though the model used a 70 kg patient, the result can be applied to any patient. To apply these result to any body weight, any of the values BLs, BLH and ANHH or PRBC given in the table need to be multiplied by the factor we will call T

$T = \frac {70 \times \text{patient's weight in kg}} {4900}$

Basically, the model considered above is designed to predict the maximum RCM that can save ANH.

In summary, the efficacy of ANH has been described mathematically by means of measurements of surgical blood loss and blood volume flow measurement. This form of analysis permits accurate estimation of the potential efficiency of the techniques and shows the application of measurement in the medical field.

Glossary of terms

ANH	Acute Normovolemic Hemodilution
ANHu	Number of Units During ANH
BLH	Maximum Blood Loss Possible When ANH Is Used Before Homologous Blood Transfusion Is Needed
BLI	Incremental Blood Loss Possible with ANH.(BLH – BLs)
BLs	Maximum blood loss without ANH before homologous blood transfusion is required
EBV	Estimated Blood Volume(70 mL/kg)
Hct	Haematocrit Always Expressed Here As A Fraction
Hi	Initial Haematocrit
Hm	Minimum Safe Haematocrit
PRBC	Packed Red Blood Cell Equivalent Saved by ANH
RCM	Red cell mass.
RCMH	Cell Mass Available For Transfusion after ANH
RCMI	Red Cell Mass Saved by ANH
SBL	Surgical Blood Loss

^[15]

References

1. Gerard J. Tortora, Bryan Derrickson (2012). "The Cardiovascular System: The Blood". Principles of Anatomy & Physiology, 13th. John Wiley & Sons, Inc. pp. 729–732. ISBN 978-0470-56510-0.
2. Joel S. Fieldman, Duong H. Phong, Yvan Saint-Aubin and Luc Vinet (2007). "Rheology". Biology and Mechanics of Blood Flows, Part II: Mechanics and Medical Aspects. Springer. pp. 119–123. ISBN 978-0-387-74848-1.
3. Gerard J. Tortora, Bryan Derrickson (2012). "The Cardiovascular System: The Blood". Principles of Anatomy & Physiology, 13th. John Wiley & Sons, Inc. pp. 729–732. ISBN 978-0470-56510-0.
4. Gerard J. Tortora, Bryan Derrickson (2012). "The Cardiovascular System: Blood Vessels and Hemodynamics". Principles of Anatomy & Physiology, 13th. John Wiley & Sons, Inc. p. 816. ISBN 978-0470-56510-0.
5. Stücker, M.; Bailer, V.; Reuther, T; Hoffman, K.; Kellam, K.; Altmeyer, P (1996). "Capillary Blood Cell Velocity in Human Skin Capillaries Located Perpendicularly to the Skin Surface: Measured by a New Laser Doppler Anemometer". Microvasc Research 52 (2): 188–192. doi:10.1006/mvre.1996.0054. PMID 8901447.
6. Gerard J. Tortora, Bryan Derrickson (2012). "The Cardiovascular System: Blood Vessels and Hemodynamics". Principles of Anatomy & Physiology, 13th. John Wiley & Sons, Inc. "Laminar flow analysis". p. 817. ISBN 978-0470-56510-0.
7. Gerard J. Tortora, Bryan Derrickson (2012). "The Cardiovascular System: Blood Vessels and Hemodynamics". Principles of Anatomy & Physiology, 13th. John Wiley & Sons, Inc. p. 816. ISBN 978-0470-56510-0.
8. Elaine N. Marieb, Katja Hoehn. (2013). "The Cardiovascular System:Blood Vessels". Human anatomy & physiology, 9th ed. Pearson Education,Inc. p. 712. ISBN 978-0-321-74326-8.
9. 'caro C.G, Pedley, T.J, Schroter R.C., Seed. W.A. (1978). The Mechanics of Circulation. Oxford University Press. pp. 3–60, 151–176. ISBN 0-19-263323-6.
10. Potters (13 February 2014). "Measuring Wall Shear Stress Using Velocity-Encoded MRI". Current Cardiovascular Imaging Reports 7. doi:10.1007/s12410-014-9257-1. Retrieved 16 September 2014.
11. Tazraei, P.; Riasi, A.; Takabi, B. (2015). "The influence of the non-Newtonian properties of blood on blood-hammer through the posterior cerebral artery". Mathematical biosciences 264: 119–127. doi:10.1016/j.mbs.2015.03.013.
12. Rand, Peter (31 May 1963). "Human blood under normothermic and hypothermic conditions" (PDF). Journal of Applied Physiology. Retrieved 16 September 2014.
13. "Bloodless medicine and surgery". Retrieved 5 April 2011.
14. "Efficacy of Acute Normovolemic hemodilution,Accessed as a Function of Blood lost". the journal of American society of anesthsiologist inc. Retrieved 5 April 2011.
15. "Hemodilution:Modelling and clinincal Aspects". IEEE. Retrieved 5 April 2011.
16. "maximum blood savings by acute Normovolemic hemodilution" (PDF). anesthesia & analgesia the gold standard in anesthesiology. Retrieved 5 April 2011.